Onchocerciasis is an endemic disease in parts of sub-Saharan Africa. Complex mathematical models are being used to assess the likely efficacy of efforts to eradicate the disease; however, their predictions have not always been borne out in practice. In this paper, we represent the immunological aspects of the disease with a single empirical parameter in order to reduce the model complexity. Asymptotic approximation allows us to reduce the vector-borne epidemiological model to a model of an infectious disease with nonlinear incidence. We then consider two versions, one with continuous treatment and a more realistic one where treatment occurs only at intervals. Thorough mathematical analysis of these models yields equilibrium solutions for the continuous case, periodic solutions for the pulsed case, and conditions for the existence of endemic disease equilibria in both cases, thereby leading to simple model criteria for eradication. The analytical results and numerical experiments show that the continuous treatment version is an excellent approximation for the pulsed version and that the current onchocerciasis eradication strategy is inadequate for regions where the incidence is highest and unacceptably slow even when the long-term behavior is the disease-free state.

J. Remme, O. Ba, K. Y. Dadzie and M. Karam, A force-of-infection model for onchocerciasis and its applications in the epidemiological evaluation of the Onchocerciasis Control
Programme in the Volta River basin area, Bulletin of the World Health Organization, 64
(1986), 667–681, URL: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2490951/.

[17]

F. Richards, N. Rizzo and A. Domínguez,
One hundred years after its discovery in Guatemala by Rodolfo Robles, Onchocerca volvulus transmission has been eliminated from the Central Endemic Zone, The American Journal Of Tropical Medicine And Hygiene [serial online], 93 (2015), 1295-1304.
doi: 10.4269/ajtmh.15-0364.

J. Remme, O. Ba, K. Y. Dadzie and M. Karam, A force-of-infection model for onchocerciasis and its applications in the epidemiological evaluation of the Onchocerciasis Control
Programme in the Volta River basin area, Bulletin of the World Health Organization, 64
(1986), 667–681, URL: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2490951/.

[17]

F. Richards, N. Rizzo and A. Domínguez,
One hundred years after its discovery in Guatemala by Rodolfo Robles, Onchocerca volvulus transmission has been eliminated from the Central Endemic Zone, The American Journal Of Tropical Medicine And Hygiene [serial online], 93 (2015), 1295-1304.
doi: 10.4269/ajtmh.15-0364.

Figure 1.
The O. volvulus life cycle. Microfilariae produced in the human host are transmitted to the black fly of the genus Simulium via a bite. Within the black fly, the larvae pass through larval stages L1-L3. At larval stage L3, they are transmitted to a human host via a bite, where they pass through the final larval stages L3-L5 and become adults [8]

Figure 2.
The SEIPMS-UV epidemiological model. Fly bites transfer the disease from infected flies ($V$) to susceptible humans ($S$) and from infective humans ($I$, $P$, and $M$) to uninfected flies ($U$), with the transfer from medicated humans ($M$) decreased by a factor $\nu$. Exposed humans ($E$) become infective, with the fraction $p$ counting as premedicated ($P$) while waiting for treatment and the remaining fraction $q = 1-p$ becoming unmedicated infectives ($I$). Premedicated humans become medicated when they receive treatment. All three infective classes can become susceptible by clearing all the adult worms. Birth and death rates for humans are equal, with all births into the susceptible class; similarly, birth and death rates for flies are equal with all births into the uninfected class